Vergauwen, A Metalogical Theory of Reference, 2.3.2 Principles of a Montague Grammar, 2.3.2.2 Structure, 2.3.2.2.1 Languages and categorical grammars

by Corry Shores
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[The following is summary. Paragraph headings are my own.]

Roger Vergauwen

A Metalogical Theory of Reference: Realism and Essentialism in Semantics

Chapter 2: Reference and Theory of Reference

2.3.2 Principles of a Montague Grammar

2.3.2.2. Structure of a Montague Grammar

Vergauwen will give a general overview of Montague's ideas.

2.3.2.2.1 Languages and categorical grammars

Montague held a view about natural languages. He believed that they contained a mechanism that produces unambiguous grammatical sentences. He called them the "Disambiguated language" (DL). The language also has a disambiguating relation R. It guarantees that the natural language's ambiguities will not appear in the syntactic metalanguage. This relation allows us to connect an ambiguous sentence to descriptions of its structure. These structural descriptions have been constructed from the syntax. We will now examine Montague's complex formulation for DL.

In DL, there is a set of all basic expressions: X. We call it the lexicon. To it we index specific syntactic categories. We consider then syntactical categories such as 'noun,' 'intransitive verb,' and so forth. We will call the whole set of all such syntactical categories upper-case delta Δ. And we will call one specific such category, like 'noun,' lower-case delta δ. We will call the set of all numbers upper-case gamma Γ. And we will call some natural number from this set lower-case gamma γ. Now we will consider a function F. In the first place, we will give it a subscript which will be a number. That number will be a number γ included in the set Γ.

Such a function regards as argument the elements from either the whole basic lexicon or from more complex categories that we form by placing the simpler categories into a specific order. The rules we use to make these formations we call the syntactic rules S. A part of these rules are the operations

We will take A to mean the set of all 'proper expressions' of a language. A basic expression that is indexed with a specific syntactic category δ we denote with:

We build up set A by repeatedly applying the operations

to the basic expressions

So in other words, we take basic elements, like 'noun,' 'article,' and 'preposition.' We use rules to create prepositional phrases, for example, 'to the dog.' We can continue building, but we might end up with such proper constructions as 'He to the dog ran.' We see that this is not a grammatical sentence or well-formed formula (WFF). We need also the syntactical rules S to extract all the grammatical sentences from the set A of proper expressions. We represent these syntactical rules as FoR. Hence we may represent DL as this quintuple:

We form the syntactic rules by means of a categorical grammar, as established by Adjukiewicz. [I quote from Vergauwen's text]

a) A finite number of basic categories are specified.

b) A set of derived categories are construed from the basic categories.

c) There are one or more syntactic rules in which the working of a syntactic operation is established and the category of the result of this operation is determined.

d) Each lexical element of the language is assigned to one syntactic category.

(25a)

In Montegue's "Proper Treatment of Quantification in Ordinary English" model, there are only two basic syntactic categories:

1) e for entity and

2) t for truth value.

We describe our whole syntax with these elements. But no words in the lexicon correspond to these categories. For example, 'to sleep.' Logically speaking, 'to sleep' is a one-place predicate. And it refers to 'the set of sleeping things.' We may also consider this predicate as a function. It would map some individual to a truth value. So if Joe is sleeping, then this predicate would map the element Joe to the truth value 'true.' So let's consider the intransitive verb (IV) 'to sleep.' Above we just said that this IV is a function that maps an individual (e) to a truth value (t). So we could formulate it as follows:

IV = t/e (a function from e to t)

The proper noun 'Joe' can be combined syntactically to an intransitive verb. We will call a proper noun E. And we want to say that when we combine a proper noun with an intransitive verb, we obtain a sentence. To do so, we formulate it as following [I cannot account for this formulation, however]

E = t/IV or t/ (t/e).

We will now articulate a syntactic rule in categorial grammar that tells how we combine proper nouns with intransitive verbs. To understand the following formulation, we will first look at its parts. We will use P to mean "phrase." And we will consider a function

that belongs to

This function will refer to the operation that joins two elements. Here it is by means of concatenation. The result is sentence

Together our rule reads [and I cannot explain it]

We can then take any sentence of our language, and formulate it unambiguously this way. This is the syntactic component of Montague's grammar. Now we consider the semantic component.

[Other Entries in this Series]

Vergauwen, Roger. A Metalogical Theory of Reference: Realism and Essentialism in Semantics. London: University Press of America, 1993.